Properties

Label 34T21
Order \(4624\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $21$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,8,12,14,15,7,3)(2,9,4,10,13,6,11,5)(18,21,31,19,30,27,34,29)(20,22,23,32,28,26,25,33), (1,25,7,21,13,34,2,30,8,26,14,22,3,18,9,31,15,27,4,23,10,19,16,32,5,28,11,24,17,20,6,33,12,29)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_8$ x 2, $C_4\times C_2$
16:  $C_8\times C_2$
136:  $C_{17}:C_{8}$ x 2
272:  34T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T21 x 7

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 17, 17 $ $16$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $16$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
$ 17, 17 $ $16$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 17 $ $16$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $16$ $17$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $16$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $8$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $16$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 17 $ $16$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $8$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $8$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $16$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
$ 17, 17 $ $16$ $17$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 17 $ $8$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $289$ $2$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 34 $ $136$ $34$ $( 1,25, 7,21,13,34, 2,30, 8,26,14,22, 3,18, 9,31,15,27, 4,23,10,19,16,32, 5, 28,11,24,17,20, 6,33,12,29)$
$ 34 $ $136$ $34$ $( 1,27,17,22,16,34,15,29,14,24,13,19,12,31,11,26,10,21, 9,33, 8,28, 7,23, 6, 18, 5,30, 4,25, 3,20, 2,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $17$ $2$ $( 1,34)( 2,22)( 3,27)( 4,32)( 5,20)( 6,25)( 7,30)( 8,18)( 9,23)(10,28)(11,33) (12,21)(13,26)(14,31)(15,19)(16,24)(17,29)$
$ 34 $ $136$ $34$ $( 1,28, 8,27,15,26, 5,25,12,24, 2,23, 9,22,16,21, 6,20,13,19, 3,18,10,34,17, 33, 7,32,14,31, 4,30,11,29)$
$ 34 $ $136$ $34$ $( 1,34, 5,31, 9,28,13,25,17,22, 4,19, 8,33,12,30,16,27, 3,24, 7,21,11,18,15, 32, 2,29, 6,26,10,23,14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $17$ $2$ $( 1,25)( 2,20)( 3,32)( 4,27)( 5,22)( 6,34)( 7,29)( 8,24)( 9,19)(10,31)(11,26) (12,21)(13,33)(14,28)(15,23)(16,18)(17,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $289$ $4$ $( 1,24, 5,29)( 2,21, 4,32)( 3,18)( 6,26,17,27)( 7,23,16,30)( 8,20,15,33) ( 9,34,14,19)(10,31,13,22)(11,28,12,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $289$ $4$ $( 1,29)( 2,32,17,26)( 3,18,16,23)( 4,21,15,20)( 5,24,14,34)( 6,27,13,31) ( 7,30,12,28)( 8,33,11,25)( 9,19,10,22)$
$ 8, 8, 8, 8, 2 $ $289$ $8$ $( 1,30,13,26,16,25, 4,29)( 2,24, 9,33,15,31, 8,22)( 3,18, 5,23,14,20,12,32) ( 6,34,10,27,11,21, 7,28)(17,19)$
$ 8, 8, 8, 8, 2 $ $289$ $8$ $( 1,23, 5,30, 6,19, 2,29)( 3,18,14,33, 4,24,10,26)( 7,25,15,22,17,34, 9,20) ( 8,31,11,32,16,28,13,27)(12,21)$
$ 8, 8, 8, 8, 2 $ $289$ $8$ $( 1,21,11,23,17,31, 7,29)( 2,28,15,34,16,24, 3,18)( 4,25, 6,22,14,27,12,30) ( 5,32,10,33,13,20, 8,19)( 9,26)$
$ 8, 8, 8, 8, 2 $ $289$ $8$ $( 1,32, 9,27, 7,24,16,29)( 2,25,13,33, 6,31,12,23)( 3,18,17,22, 5,21, 8,34) ( 4,28)(10,20,11,30,15,19,14,26)$

Group invariants

Order:  $4624=2^{4} \cdot 17^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.